As part of a larger article, I am busy writing a shorthand derivation / evaluation of the expected value of a random variable $X$:
$\mathop{{}\mathbb{E^Q}}[f(X)] = \mathop{{}\int_{-{\infty}}^{\infty}}f(X) \cdot d \mathbb{Q}(S)$
Here $\mathbb{Q}$ is a probability measure and $S$ is distributed lognormally with $F_S(x)$ the cumulative distribution function of $S$.
From the reference articles that I am using, I deduce that that the following is true:
$\begin{aligned} \mathop{{}\mathbb{E^Q}}[f(X)] &= \mathop{{}\int_{-{\infty}}^{\infty}}f(X) \cdot d \mathbb{Q}(S) \\ &= \mathop{{}\int_{-{\infty}}^{\infty}}f(X) \cdot d {F}(S) \\ \end{aligned}$
In other words, loosely speaking, we are saying that $\mathbb{Q} \approx F$.
Question:
I am seeking the correct mathematical phrasing that justifies equating / substituting $d \mathbb{Q}$ with $d F$
EDIT:
Below is a screenshot from the source reference to help explain what I am trying to achieve / understand. To address the comments; firstly thank you. Secondly I have never heard of the pushforward measure and on that basis I doubt it is related, but I am not an expert on measure theory by any means of the word so it may very well be related.
Here $S$ represents the stochastic process followed by a stock / share in a company:
$ dS = \mu S dt + \sigma S d {W}_t $
where $dW_t$ is a Wiener process indexed by time $t$ and $ \sim N(0,t) $.
As mentioned briefly, $S$ is distributed lognormally with mean and variance as per the screenshot below.
In the article (see below) they start with the expected value under some measure $Q$. My understanding is that the general representation of expected values (using Lebesgue notation) is
$\begin{aligned} \mathop{{}\mathbb{E^Q}}[X] = \mathop{{}\int X \cdot dQ} \end{aligned}$
In the article however, they skip the $dQ$ step and start with $dF(S)$ with $F$ the cdf of $S$.
Goal: I would like to properly explain how $Q$ and $F$ are "interchangeable".
